Hollow workpieces having thin walled cylindrical or tubular wall surfaces to be machined are easily distortable and are best supported for machining by specialized chuck apparatus, such as the type disclosed in the present assigneels U.S. Pat. No. 4,953,877 to Slachta et al, granted Sep. 4, 1990. The Slachta et al chucking device employs a plurality of discrete flexible chucking fingers actuated by a pliable inflatable ring for radially deflecting the fingers into gripping, non-distorting engagement with the peripheral wall of the workpiece. Some workpiece walls are concentric and can be gripped without repositioning them. Other workpieces will be gripped eccentrically without relocating them. In machining many other workpieces, it has been found necessary to first accurately relocate the peripheral wall of the workpiece in a position which is concentric with the axis of the chuck prior to fully gripping it, and a chuck of the general type disclosed in the Slachta et al patent has been improved to achieve the particular result desired.